Thank you so much! Im tutoring for some of my classmates and i realized i dont know how to explain this concept, because I just memorized that its reciprocated myself.
Students do often struggle to grasp this opposite effect, so I try to showing them in many ways. I use the "x had to be 2 more to get the same y" explanation. I also show how if you were to solve for x instead of y then the horizontal ones would look as you expect and the vertical would be opposite. And how that relates to both of then being opposite in something like a circle or ellipse equation. But this is a bit too much at once for students to process, so the quicker answer is what you did in this video. I find it's actually more difficult to get students to fully understand how the order of multiple [especially horizontal] transformations work. Many teachers seem to tell their students to "always do the reflections and shrinks/stretches first and then the translations" (or at least this is the impression students get fron what the teacher says), but this isn't fully correct. It relies on the context of this being a problem where you are looking at a graph and wanting to write the equation yourself in the easiest-to-understand way. Students [more often than not] do not realize that the order of transformations matches the order of operations for vertical transformations and is the opposite for horizontal transformations. I like to view the order for horizontal transformations as the same order that you'd solve for x if you set the stuff inside the parent function equal to zero, and this also bolsters the intuition of each horizontal transformation needing an "opposite" interpretation. So, long story short, even a student who understands y = sqrt(2x) is a horizontal SHRINK and y = sqrt(x+1) is a translation LEFT may not realize that y = sqrt(2x+1) would go left FIRST and then do the horizontal shrink AFTERWARD. You can show the student why this must be the case, but they often don't even pause to consider that they need to deduce the correct order of transformations, since they're relying on the "stretches/shrinks always come before translations" slight misunderstanding. Of course, one could rewrite y = sqrt(2x+1) as y = sqrt(2(x+1/2)) in order to have the horizontal shrink come first and then do a smaller translation left, but (1) the student usually won't think to rewrite it & (2) a teacher may want the transformations interpretted "as-is" instead of based on a rewriting of the equation. Do you have any thoughts on this? I admit thankfully it doesn't come up much, as teachers and textbooks don't often give multiple horizontal transformations within the same problem (and when they do, there's still at least a 50% chance they'll match the "stretches/shrinks first" approach).
But why do you have to left one first then multiply 1/2 to the x values? I tried graphing the “wrong way” by doing *1/2 first then subtract 1 and I understand how it messes the graph up and that the graph is not right according to its values . But why does it happen? Why -1 then *1/2? I thought you’re supposed to go by order of operations but suddenly its now the opposite..
@@davidchanpadid1729 The original function is y = sqrt(x) and the transformed function is y = sqrt(2x+1). However, if you solve each of these equations for x, you get that the original equation is the same as x = y^2 (with a restriction that y >= 0). And the transformed equation is the same as x = (y^2 - 1) / 2. Try it yourself and see. From this perspective, order of operations IS being followed. We handle the "minus one" first to decrease all x coordinates by 1 (shifting left) and then the "divide by 2" to multiply all x coordinates by 1/2 (horizontally shrinking). So, you see, our vertical transformations follow order of operations when our equation is of the form "y= something involving x", and our horizontal transformations equally follow order of operations when our equation is of the form "x= something involving y". It is "moving a variable to the complicated side of the equation" that results in all the operations and their order being reversed. You could even do this with vertical transformations. Consider y = 3x^3 + 5. Clearly, we vertically stretch by 3 first and then we translate up 5 units. However, if I solve that equation for x, I must first do the opposite of the FINAL operation being done - that is, I must subtract 5 to get rid of the +5. Then I would have y - 5 = 3x^3. After that, I'd divide both sides by 3 to cancel the "times 3", giving me (y-5)/3 = x^3. Then cube root. So, you see, the operation that had happened earlier by order of operations in y = 3x^3 + 5 is having its opposite happen later by order of operations now in x = cbrt((y-5)/3). You can see hints at this phenomenon when you consider things like point-slope form where we write y - y1 = m(x-x1). That "minus y1" on the left side is moving the function up so it passes through (x1, y1) instead of through (x1, 0). You can see it with ellipses too. When we write (x-h)^2 / a^2 + (y-k)^2 / b^2 = 1, we can instead undistribute the square over the fractions to get ((x-h)/a)^2 + ((y-k)/b)^2 = 1. Since both x and y are on the complicated side of the equation, they both get the opposite transformation behavior using the operations beside them. Specifically, we start with the parent graph x^2 + y^2 = 1, a circle of radius 1 centered at the origin. The horizontal stretch by "a" happens before doing the translation right h units that gets the ellipse to be centered on the correct x-value. And the vertical stretch by "b" happens before going up k units. Ultimately, this gives us an ellipse with "x-radius" a and "y-radius" b with a center at (h,k). If you already understand this to be the equation of such an ellipse, then, in a sense, you understand that we need to use reverse order of operations to interpret transformations when x and/or y is on the complicated side of the equation. I possibly said too much and maybe should've made this comment shorter. But I hope something here was helpful. Keep doing examples, and you'll likely grow to understand it in time.
@@davidchanpadid1729 Fwiw, I did reply, but it looks like RUclips automatically deleted it. 🤷♂️ So much for all that typing I did. Or there's a display error and it'll be back later. In short, the reason for the reverse order of operations is that you have to move the last thing by order of operations over to the other side of the equation FIRST when solving for x. So, even though it's reverse order of operations now, it would be normal order of operations if you were to solve for x.
@@Dreamprism ohh.. So you are really just solving for x thats why you do it in inverse right? Ex: sqrt(2x+1) , (FIRST WAY) To get a value of 1 you have to solve for x by subtracting 1 first then dividing 2. (SECOND WAY) Or you could also divide 2 first but you would have to subtract 1/2 which would still give you x=0. So.. you could do sqrt(2x+1) but have to do the FIRST WAY . Or (like youve mentioned) can rewrite as sqrt[2(x+1/2)] but you have to do it the SECOND WAY … Am I getting the right idea here ? And man thank you so much for taking the time to reply. I’m grateful for you🙏 .. I luckily copied this text first before sending cuz my comment also disappeared lol. make sure you copy too if you reply again.. youtube censoring comments for no reason
@@davidchanpadid1729 Yeah, the order of transformations is consistent with the order you'd move stuff to the other side to solve for x, which is the reverse order of operations to how it's originally expressed. You have it right.
I was actually thinking about this yesterday drawing some graphs with functions! now correct me if i am wrong but it seems that transformations on the same axis as the variable have the opposite effect, that's why when you add the +K value (shift on the y axis) to the radical function the positive raises it up, same effect unlike the shift on the x axis, now try this out go and graph a normal unit circle function then try to raise the unit circle up the y axis. in order to do that you actually need to add -1 instead of 1. its like how Mario explained it in order to get the same value you need to put in the opposite effect, also if you think about it the shift on the y axis if you move it to the other side of the function then it would be Y-K so yeah its interesting to think about, Hopefully i didn't get anything wrong in this comment, also great video as usual!!
Thank you so much! Im tutoring for some of my classmates and i realized i dont know how to explain this concept, because I just memorized that its reciprocated myself.
Students do often struggle to grasp this opposite effect, so I try to showing them in many ways. I use the "x had to be 2 more to get the same y" explanation.
I also show how if you were to solve for x instead of y then the horizontal ones would look as you expect and the vertical would be opposite. And how that relates to both of then being opposite in something like a circle or ellipse equation. But this is a bit too much at once for students to process, so the quicker answer is what you did in this video.
I find it's actually more difficult to get students to fully understand how the order of multiple [especially horizontal] transformations work.
Many teachers seem to tell their students to "always do the reflections and shrinks/stretches first and then the translations" (or at least this is the impression students get fron what the teacher says), but this isn't fully correct. It relies on the context of this being a problem where you are looking at a graph and wanting to write the equation yourself in the easiest-to-understand way.
Students [more often than not] do not realize that the order of transformations matches the order of operations for vertical transformations and is the opposite for horizontal transformations.
I like to view the order for horizontal transformations as the same order that you'd solve for x if you set the stuff inside the parent function equal to zero, and this also bolsters the intuition of each horizontal transformation needing an "opposite" interpretation.
So, long story short, even a student who understands y = sqrt(2x) is a horizontal SHRINK and y = sqrt(x+1) is a translation LEFT may not realize that y = sqrt(2x+1) would go left FIRST and then do the horizontal shrink AFTERWARD. You can show the student why this must be the case, but they often don't even pause to consider that they need to deduce the correct order of transformations, since they're relying on the "stretches/shrinks always come before translations" slight misunderstanding.
Of course, one could rewrite y = sqrt(2x+1) as y = sqrt(2(x+1/2)) in order to have the horizontal shrink come first and then do a smaller translation left, but (1) the student usually won't think to rewrite it & (2) a teacher may want the transformations interpretted "as-is" instead of based on a rewriting of the equation.
Do you have any thoughts on this? I admit thankfully it doesn't come up much, as teachers and textbooks don't often give multiple horizontal transformations within the same problem (and when they do, there's still at least a 50% chance they'll match the "stretches/shrinks first" approach).
But why do you have to left one first then multiply 1/2 to the x values? I tried graphing the “wrong way” by doing *1/2 first then subtract 1 and I understand how it messes the graph up and that the graph is not right according to its values . But why does it happen? Why -1 then *1/2? I thought you’re supposed to go by order of operations but suddenly its now the opposite..
@@davidchanpadid1729 The original function is y = sqrt(x) and the transformed function is y = sqrt(2x+1).
However, if you solve each of these equations for x, you get that the original equation is the same as x = y^2 (with a restriction that y >= 0). And the transformed equation is the same as x = (y^2 - 1) / 2. Try it yourself and see.
From this perspective, order of operations IS being followed. We handle the "minus one" first to decrease all x coordinates by 1 (shifting left) and then the "divide by 2" to multiply all x coordinates by 1/2 (horizontally shrinking).
So, you see, our vertical transformations follow order of operations when our equation is of the form "y= something involving x", and our horizontal transformations equally follow order of operations when our equation is of the form "x= something involving y".
It is "moving a variable to the complicated side of the equation" that results in all the operations and their order being reversed.
You could even do this with vertical transformations. Consider y = 3x^3 + 5. Clearly, we vertically stretch by 3 first and then we translate up 5 units.
However, if I solve that equation for x, I must first do the opposite of the FINAL operation being done - that is, I must subtract 5 to get rid of the +5. Then I would have y - 5 = 3x^3. After that, I'd divide both sides by 3 to cancel the "times 3", giving me (y-5)/3 = x^3. Then cube root.
So, you see, the operation that had happened earlier by order of operations in y = 3x^3 + 5 is having its opposite happen later by order of operations now in x = cbrt((y-5)/3).
You can see hints at this phenomenon when you consider things like point-slope form where we write y - y1 = m(x-x1). That "minus y1" on the left side is moving the function up so it passes through (x1, y1) instead of through (x1, 0).
You can see it with ellipses too. When we write (x-h)^2 / a^2 + (y-k)^2 / b^2 = 1, we can instead undistribute the square over the fractions to get ((x-h)/a)^2 + ((y-k)/b)^2 = 1. Since both x and y are on the complicated side of the equation, they both get the opposite transformation behavior using the operations beside them.
Specifically, we start with the parent graph x^2 + y^2 = 1, a circle of radius 1 centered at the origin. The horizontal stretch by "a" happens before doing the translation right h units that gets the ellipse to be centered on the correct x-value. And the vertical stretch by "b" happens before going up k units. Ultimately, this gives us an ellipse with "x-radius" a and "y-radius" b with a center at (h,k).
If you already understand this to be the equation of such an ellipse, then, in a sense, you understand that we need to use reverse order of operations to interpret transformations when x and/or y is on the complicated side of the equation.
I possibly said too much and maybe should've made this comment shorter. But I hope something here was helpful. Keep doing examples, and you'll likely grow to understand it in time.
@@davidchanpadid1729 Fwiw, I did reply, but it looks like RUclips automatically deleted it. 🤷♂️ So much for all that typing I did.
Or there's a display error and it'll be back later.
In short, the reason for the reverse order of operations is that you have to move the last thing by order of operations over to the other side of the equation FIRST when solving for x. So, even though it's reverse order of operations now, it would be normal order of operations if you were to solve for x.
@@Dreamprism ohh.. So you are really just solving for x thats why you do it in inverse right? Ex: sqrt(2x+1) , (FIRST WAY) To get a value of 1 you have to solve for x by subtracting 1 first then dividing 2. (SECOND WAY) Or you could also divide 2 first but you would have to subtract 1/2 which would still give you x=0. So.. you could do sqrt(2x+1) but have to do the FIRST WAY . Or (like youve mentioned) can rewrite as sqrt[2(x+1/2)] but you have to do it the SECOND WAY … Am I getting the right idea here ? And man thank you so much for taking the time to reply. I’m grateful for you🙏 .. I luckily copied this text first before sending cuz my comment also disappeared lol. make sure you copy too if you reply again.. youtube censoring comments for no reason
@@davidchanpadid1729 Yeah, the order of transformations is consistent with the order you'd move stuff to the other side to solve for x, which is the reverse order of operations to how it's originally expressed.
You have it right.
I was actually thinking about this yesterday drawing some graphs with functions! now correct me if i am wrong but it seems that transformations on the same axis as the variable have the opposite effect, that's why when you add the +K value (shift on the y axis) to the radical function the positive raises it up, same effect unlike the shift on the x axis, now try this out go and graph a normal unit circle function then try to raise the unit circle up the y axis. in order to do that you actually need to add -1 instead of 1. its like how Mario explained it in order to get the same value you need to put in the opposite effect, also if you think about it the shift on the y axis if you move it to the other side of the function then it would be Y-K so yeah its interesting to think about, Hopefully i didn't get anything wrong in this comment, also great video as usual!!
thank you for your help😭😭❤️
You’re welcome!